Research On Double-input Single-output Fuzzy Systems And Their Modeling Problems

In the field of fuzzy theory and fuzzy control, the modeling of fuzzy systems and their universal approximation are the important research objectives. With double input single output (DISO) fuzzy systems as the study objectives and construction of fuzzy systems as the research target, this paper investigates some existing problems during the modeling of DISO fuzzy systems, and several types of DISO fuzzy system are constructed. The simulation results also illustrate the feasibility and effective of the constructed fuzzy systems. The main research of this paper is as followed:Firstly, to figure out the problem that the constructed fuzzy systems based on singleton fuzzifiers and normal fuzzy implications are not universal approximations, the method of non-singleton fuzzifier is popularized in this paper and four DISO fuzzy systems are built. Through fuzzy transformation, the constructed fuzzy systems are proved to be universal approximators. And the simulation results show that the constructed fuzzy systems can reach high approximation precision. The research indicates that we are able to construct fuzzy systems with the capability of approximation based on normal fuzzy implications.Secondly, the constructed fuzzy systems are applied to solve ordinary differential equations. As is known to all, numerical solutions could be obtained merely for most complex differential equations. With regard to this issue, we first acquire the numerical solutions of ordinary differential equations. Using these numerical solutions as the input-output information of certain system, a DISO fuzzy system is constructed and its capability of approximation is demonstrated. Then different types of ordinary differential equation are tested with or without interference and simulations reveal that the approximation error between the constructed system and the theoretical solution is small. Consequently, the constructed fuzzy system could serve as approximate analytical solution of ordinary differential equations.Finally, the modeling of fuzzy systems is studied. By the present approaches of modeling, complex non-linear differential equations in each interval are obtained, which are not convenient for application. To solve this problem, this paper proposes a method of modeling based on DISO fuzzy systems. By this method, linear differential equations in each interval could be achieved. So this method overcomes the construction complexity of linear models and the decline of approximation precision caused by linear edge. Then the linear model is proved to be a universal approximator and is compared with the nonlinear model through simulation. The simulation results indicate that the approximation precision of the linear model is rather high, even though lower than that of the non-linear model. Then we obtain the differential equation model of the linear model we construct by the method of fuzzy inference. Through the simulation of several typical double-input single-output systems, the linear differential equation model is proved to be of great approximation accuracy.